Question: $y'=5y$ Is $y=3e^{x^5}$ a solution to the above equation? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Answer: In order to find whether $y=3e^{x^5}$ is a solution, we need to substitute it into the equation and see if we get equivalent expressions on each side of the equal sign. In addition to substituting for $y$, we need to find the corresponding $y'$ expression to substitute into the equation: $\begin{aligned} y'&=\dfrac{d}{dx}\left[3e^{x^5}\right] \\\\ &=15x^4e^{x^5} \end{aligned}$ Now we substitute ${y=3e^{x^5}}$ and ${y'=15x^4e^{x^5}}$ into the equation: $\begin{aligned} {y'}&=5{y} \\\\ {15x^4e^{x^5}}&\stackrel{?}{=}5\left({3e^{x^5}}\right) \\\\ 15x^4e^{x^5}&\stackrel{?}{=} 15e^{x^5} \\\\ x^4&\neq 1 \end{aligned}$ We did not obtain equivalent expressions on each side. In conclusion, no, $y=3e^{x^5}$ is not a solution to the differential equation.